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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2021
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2108.05254 |
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| _version_ | 1866910405507416064 |
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| author | Bressan, Alberto Galtung, Sondre T. Sun, Qing |
| author_facet | Bressan, Alberto Galtung, Sondre T. Sun, Qing |
| contents | The paper studies a class of variational problems, modeling optimal shapes for tree roots. Given a measure $μ$ describing the distribution of root hair cells, we seek to maximize a harvest functional $\mathcal{H}$, computing the total amount of water and nutrients gathered by the roots, subject to a cost for transporting these nutrients from the roots to the trunk. Earlier papers had established the existence of an optimal measure, and a priori bounds. Here we derive necessary conditions for optimality. Moreover, in space dimension $d=2$, we prove that the support of an optimal measure is nowhere dense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_05254 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Optimal Shapes for Tree Roots Bressan, Alberto Galtung, Sondre T. Sun, Qing Optimization and Control 35R06, 49Q10, 90B06, 92C80 The paper studies a class of variational problems, modeling optimal shapes for tree roots. Given a measure $μ$ describing the distribution of root hair cells, we seek to maximize a harvest functional $\mathcal{H}$, computing the total amount of water and nutrients gathered by the roots, subject to a cost for transporting these nutrients from the roots to the trunk. Earlier papers had established the existence of an optimal measure, and a priori bounds. Here we derive necessary conditions for optimality. Moreover, in space dimension $d=2$, we prove that the support of an optimal measure is nowhere dense. |
| title | Optimal Shapes for Tree Roots |
| topic | Optimization and Control 35R06, 49Q10, 90B06, 92C80 |
| url | https://arxiv.org/abs/2108.05254 |